Two situations could lend support to the above scenario. Firstly, we
consider the case with a reversal of the direction of the magnetic
field, i.e., a magnetic field in the \(-z\) -direction. From the above
argument, the polarizations of the induced magnetic orders should be
reversed if the magnetic field reverses its direction. It is exactly
the case as evidenced in Figs. REF (a) and REF (b),
where the results are obtained with an out-of-plane magnetic field
in the \(-z\) -direction while keep other parameters the same as that
in Fig. REF . Secondly, we should make a comparison with the
spin-orbital coupling (SOC) term in Kane-Mele model [1]},
which has the form
\(H_{KM}=i\beta \sum _{i,\mathbf {\tau }^{\prime }_{j},\sigma ,\sigma ^{\prime }}\hat{\sigma }^{z}_{\sigma \sigma ^{\prime }}(-1)^{j}(a^{\dag }_{i,\sigma }a_{i+\mathbf {\tau }^{\prime }_{j},\sigma ^{\prime }}-b^{\dag }_{i,\sigma }b_{i+\mathbf {\tau }^{\prime }_{j},\sigma ^{\prime }}).\) 
